Find the maximum and minimal values of the function
$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$
over its domain.
Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.
Determine if the following infinite series is convergent or divergent:
$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$
Show that $\ln x < \sqrt{x}$ holds for all positive $x$.
Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$
Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.
Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(2)+f(\frac{1}{2})$.
Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$
Compute
$$\int_0^{\infty}\frac{x^2}{1+x^4}dx$$
Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where
$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$
Determine the values of $\alpha$ and $\beta$ such that
$$\lim_{n\to\infty}\frac{n^{\alpha}}{n^{\beta}-(n-1)^{\beta}}=2020$$
Evaluate
$$\int_0^1 x\arcsin{x}d{x}$$
Evaluate
$$\int_0^1 \sqrt{1-x^2} d{x}$$
Compute
$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$
where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.
Compute $$\int \ln{x} dx$$
Which one of the numbers below is larger?
$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that
$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$
Prove the absolute convergence testing rule using the comparison testing rule. That is, if a series $\{|a_n|\}$ converges, then the series $\{a_n\}$ must be convergent.
For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?
$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$
Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by
$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$
achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.
For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.
Let $s\in\mathbb{R}$. Prove that
$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$
converges if and only if $s > 1$.
A right circular cone $\mathbb{C}$ has altitude $40$ and a circular base
of radius $30$ inches. A sphere $\mathbb{S}$ is inscribed in $\mathbb{C}$. Compute the volume of the region inside $\mathbb{C}$ which is above $\mathbb{S}$.